Computational Solid State Physics ??????? ?7?

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Computational Solid State Physics ??????? ?7?

• 7. Many-body effect I
• Hartree approximation, Hartree-Fock approximation
  andDensity functional method

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Hartree approximation
N-electron Hamiltonian
N-electron wave function
i-th spin-orbit
ortho-normal set
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Expectation value of the energy
single electron energy
Hartree interaction
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Charge density
charge density operator
charge density
Hartree interaction
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Hartree calculation for Ngtgt1
Energy minimization with condition
Self-consistent Schröedinger equation for the
i-th state
Electrostatic potential energy caused by
electron-electron Coulomb interaction
charge density
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Hartree-Fock approximation

• Pauli principle
• Identical particles
• Slater determinant
• Exchange interaction
• Hartree-Fock-Roothaans equation

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Many electron Hamiltonian
single electron Hamiltonian
electron-electron Coulomb interaction
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Slater determinant
or
N-electron wave function
John Slater
spin orbit
Permutation of N numbers
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Properties of Slater determinant
or
If
Pauli principle
Identical Fermi particles
The Slater determinant satisfies both
requirements of Pauli principle and identical
Fermi particles on N-electron wave function.
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Ground state energy
Permutation of N numbers
Orthonormal set
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Expectation value of Hamiltonian
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Expectation value of Hamiltonian
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Expectation value of many-electron Hamiltonian
Coulomb integral
Exchange integral
Hartree term between like spin electrons and
between unlike spin electrons
Fock term between like spin electrons
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Exchange interaction
Pauli principle
X
no transfer
transfer
suppression of electron-electron Coulomb energy
No suppression of electron-electron Coulomb energy
gain of exchange energy
No exchange energy
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Hartree-Fock calculation (1)
Expansion by base functions
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Hartree-Fock calculation (2)
Calculation of the expectation value
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Hartree-Fock calculation (3)
Expectation value of N-electron Hamiltonian
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Hartree-Fock calculation (4)
Minimization of E with condition
Hartree-Fock-Roothaans equation
Exchange interaction is also considered in
addition to electrostatic interaction.
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Hartree-Fock calculation (5)
Schröedinger equation for k-th state
m number of base functions N number of
electrons
Self-consistent solution on C and P
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Density functional theory

• Density functional method to calculate the ground
  state of many electrons
• Kohn-Sham equations to calculate the single
  particle state
• Flow chart of solving Kohn-Sham equation

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Many-electron Hamiltonian
T kinetic energy operator Vee electron-electron
Coulomb interaction vext external potential
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Variational principles

• Variational principle on the ground state energy
  functional En The ground state energy EGS is
  the lowest limit of En.
• Representability of the ground state energy.

charge density
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Density-functional theory

• Kohn-Sham total-energy functional for a set of
  doubly occupied electronic states

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Kohn-Sham equations
Hartree potential of the electron charge
density
exchange-correlation potential
excahnge-correlation functional
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Kohn-Sham eigenvalues

Kinetic energy functional
Janaks theorem
If f dependence of ei is small, ei means an
ionization energy.
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Local density approximation
nX(r12)
Exchange-correlation energy per electron in
homogeneous electron gas
exchange hole distribution for like spin
Local-density approximation satisfies the sum
rule.
Sum Rule
exchange-correlation hole
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Blochs theorem for periodic system
G Reciprocal lattice vector a Lattice
vector
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Plane wave representation of Kohn-Sham equations
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Supercell geometry
Point defect
Surface
Molecule
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Flow chart describing the computational procedure
for the total energy calculation
Conjugate gradient method
Molecular-dynamics method
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Hellman-Feynman force on ions (1)

• for eigenfunctions

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Hellman-Feynman force on ions (2)
Electrostatic force between an ion and electron
charge density
Electrostatic force between ions
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Problems 7

• Derive the single-electron Schröedinger equations
  in Hartree approximation.
• Derive the single-electron Schröedinger equations
  in Hartree-Fock approximation.
• Derive the Kohn-Sham equation in density
  functional method.
• Solve the sub-band structure at the interface of
  the GaAs active channel in a HEMT structure in
  Hartree approximation.